Page 64 - Artificial Intelligence for Computational Modeling of the Heart

34 Chapter 1 Multi-scale models of the heart for patient-specific simulations

for increased risk of hemolisis [166], as well as elevated transvalvu-
lar pressure drop, with increase of ventricular afterload and in-
creased risk for ventricular remodeling. On the other extreme of
the spectrum, blood can stagnate in areas of the pathologically
dilated ventricles, potentially leading to thrombus formation and
increased risk of embolism [167]. Blood is a connective tissue with
a particular rheological behavior, which makes it particularly im-
portant to account for what happens at the smaller spatial and
temporal scales of hemodynamics.
Intra-cardiac hemodynamics is intrinsically a multi-physics
problem, since the mechanical interaction of blood and myocar-
dial walls is a major determinant of cardiac flow. Fluid-structure
interaction (FSI) models are therefore well suited to describe the
coupled system [168–176]. In a general formulation, such models
describe the joint dynamics of two bodies (one solid, one fluid)
that share an interface where they mutually exchange mechani-
cal constraints. In the study of biological systems, a very popular
model for blood dynamics is based on the Navier–Stokes equa-
tions for a Newtonian fluid [177]. As discussed in Section 1.3,my-
ocardium can be modeled as a complex solid material with an ac-
tive and a passive behavior; other parts of the heart such as valves
and other connective tissues can be described as purely passive
visco-elastic materials. The solid is free to move in space under the
mechanical action of the fluid, as well as other constraints and ac-
cording to its active behavior. The fluid changes its configuration
according to its boundary conditions, which include the motion of
the fluid-solid interface. At the fluid-solid interface, blood moves
together with the structures it touches (no-slip condition). Me-
chanical stresses are transferred seamlessly across the interface.
In this section we review commonly used modeling approaches
for the detailed study of intra-cardiac hemodynamics, including
the mechanical coupling of blood with the surrounding tissues.

1.4.2.1 Modeling intra-cardiac blood flow
The Navier–Stokes equations governing the dynamics of a fluid
read:

∂u
ρ + u ·∇u = divT + b, (1.21)
∂t
div u = 0, (1.22)

where u is the fluid velocity, ρ its density, T is the stress tensor and
b is a vector field of bodily forces. The first equation expresses the
conservation of momentum, in the form of the balance between
the acceleration of the fluid on the left hand side and the total